General stability result of a thermoelastic laminated beam with Coleman-Gurtin law for the heat flux

We consider a thermoelastic laminated beam where the heat flux is given by the Coleman-Gurtin’s law. That is

$$\begin{aligned} \tau q(t)+(1-\alpha )\theta _{x}+\alpha \int _{0}^{\infty } \Psi (s)\theta _{x}(x, t-s)ds=0,\qquad \alpha \in (0, 1), \end{aligned}$$

where \(\theta \) is the temperature supposed to be known for negative times. \(\Psi \) is the convolution thermal kernel, nonnegative bounded, and convex summable function on \([0, \infty )\) and belongs to a wide class of relaxation function satisfies the unitary total mass and some additional properties that will be specified in the paper. By using the well-known Dafermos history framework and constructing a suitable Lyapunov functional, we establish general decay results for which the exponential and polynomial decay rates are special cases, depending on some assumptions on the relaxation function and the wave speeds of the system. The result obtained is new and substantially improves earlier results in the literature.